download the script: Definition of Entropy
Since the concept “reversibility” plays a crucial role in the thermodynamics, it is necessary for us to find an approach to determine whether the process reversible or not. And the most well-known Clausius Inequality provides us this possibility.
As we all know, in thermodynamics, if the cyclic integral of a parameter x is zero, that means:then this parameter x is a property of state and path-independent as well.
The Clausius Inequality indicates that for a reversible cycle process, we obtain an equation expressed as: As mentioned above, since the cyclic integral of quantity (δqrev/T) is zero, this term also represents a property of state.
Clausius discovered therefore a new thermodynamic property of state which is called “Entropy” and written as S: (This definition is the macroscopic definition of entropy. For the engineer, we usually use this definition instead of microscopic definition)
- Entropy S is an extensive property of a system and has the unit of [S]=J/K
- Entropy per unit mass s is an intensive property of a system and has the unit of [S]=J/kg·K
Since entropy is the property of state and is a point function as well, we can determine the entropy change of a system which undergoes a process (reversible or irreversible) from state 1 to state 2 as:
We notice that the entropy change can be calculated only if the system undergoes a reversible process. But luckily, the entropy is defined as a property of state and the entropy change only depends on the initial state and the final state, therefore, even though a system undergoes an irreversible process from state 1 to state 2, we can arbitrarily draw any other convenient imaginary reversible paths from 1 to 2 to calculate the entropy change.
But for an irreversible process, ds≠δq/T. According to the Clausius Inequality for an irreversible process, we have: Now assume that the system goes along a cycle process which consists of one irreversible process 1→2 (path L) and one reversible process 2→1 (path M). Therefore:
- δqrev means heat transferred during an irreversible process (path M)
- δqirr means heat transferred during a reversible process (path L)
Since 2→1 is reversible, the second term on the left side of the above inequality is:
Hence we now obtain:
We can now draw a conclusion: for an irreversible process, the entropy change of a closed system from state 1 to state 2 is greater than the integral of (δqirr/T)
The difference of Δs and (δqirr/T) ascribes to the entropy generation. (The entropy generation is discussed here: https://thermodynamics-engineer.com/the-increase-of-entropy-change/)